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Asymptotic behavior for the discrete in time heat equation

In this paper we investigate the asymptotic behavior and decay of the solution of the discrete in time $N$-dimensional heat equation. We give a convergence rate with which the solution tends to the discrete fundamental solution, and the asymptotic decay, both in $L^p(\R^N).$ Furthermore we prove optimal $L^2$-decay of solutions. Since the technique of energy methods is not applicable, we follow the approach of estimates based on the discrete fundamental solution which is given by an original integral representation and also by MacDonald's special functions. As a consequence, the analysis is different to the continuous in time heat equation and the calculations are rather involved.